Question

By considering \(\langle h \mid h\rangle\), where \(h=f+\lambda g\) with \(\lambda\) real, prove that, for two functions \(f\) and \(g\) $$ \langle f \mid f\rangle\langle g \mid g\rangle \geq \frac{1}{4}[\langle f \mid g\rangle+\langle g \mid f\rangle]^{2} $$ The function \(y(x)\) is real and positive for all \(x\). Its Fourier cosine transform \(\tilde{y}_{\mathrm{c}}(k)\) is defined by $$ \tilde{y}_{\mathrm{c}}(k)=\int_{-\infty}^{\infty} y(x) \cos (k x) d x $$ and it is given that \(\tilde{y}_{\mathrm{c}}(0)=1\). Prove that $$ \tilde{y}_{\mathrm{c}}(2 k) \geq 2\left[\tilde{y}_{\mathrm{c}}(k)\right]^{2}-1 $$.

Short answer

Using the positivity of the inner product \langle h \mid h\rangle and applying the Cauchy-Schwarz inequality for Fourier transforms proves the required inequalities.

Step-by-step solution

- Express \langle h \mid h\rangle in terms of \langle f \mid f\rangle, \langle g \mid g\rangle, and \langle f \mid g\rangle

We start by expressing \langle h \mid h\rangle, where \h = f + \lambda g\. Using the definition of the inner product, we get:\[\langle h \mid h\rangle = \langle f + \lambda g \mid f + \lambda g\rangle = \langle f \mid f\rangle + \lambda \langle f \mid g\rangle + \lambda \langle g \mid f\rangle + \lambda^2 \langle g \mid g\rangle.\]

- Use the positivity of the inner product \langle h \mid h\rangle

Since the inner product \langle h \mid h\rangle must always be non-negative, we can write:\[\langle f \mid f\rangle + \lambda \langle f \mid g\rangle + \lambda \langle g \mid f\rangle + \lambda^2 \langle g \mid g\rangle \geq 0.\]

- Find the discriminant to ensure non-negativity for all \lambda

For this quadratic expression in \lambda to be non-negative for all \lambda, its discriminant must be less than or equal to zero. The discriminant of \langle f \mid f\rangle + \lambda \langle f \mid g\rangle + \lambda \langle g \mid f\rangle + \lambda^2 \langle g \mid g\rangle\ is:\[(\langle f \mid g\rangle + \langle g \mid f\rangle)^2 - 4 \langle f \mid f\rangle \langle g \mid g\rangle \leq 0.\]

- Rearrange the inequality

Rearranging the discriminant inequality gives us:\[(\langle f \mid g\rangle + \langle g \mid f\rangle)^2 \leq 4 \langle f \mid f\rangle \langle g \mid g\rangle.\]

- Final inequality

Taking square roots on both sides yields:\[\langle f \mid f\rangle \langle g \mid g\rangle \geq \frac{1}{4}[\langle f \mid g\rangle + \langle g \mid f\rangle]^2.\]

- Define the Fourier cosine transform

Given the function \y(x) is real and positive for all \x\, its Fourier cosine transform is defined as:\[\tilde{y}_{\mathrm{c}}(k) = \int_{-\infty}^{\infty} y(x) \cos (k x) d x.\]Furthermore, it is given that \tilde{y}_{\mathrm{c}}(0) = 1\.

- Apply the Cauchy-Schwarz inequality in the context of Fourier transforms

Consider the positivity condition \tilde{y}_{\mathrm{c}}(0) = 1\ implies:\[\left[\tilde{y}_{\mathrm{c}}(k)\right]^2 \leq \tilde{y}_{\mathrm{c}}(0) \tilde{y}_{\mathrm{c}}(2k).\]

- Prove the required inequality using the given condition

Rewriting this, we aim to show:\[\tilde{y}_{\mathrm{c}}(2 k) \geq 2\left[\tilde{y}_{\mathrm{c}}(k)\right]^2 - 1.\]Since \tilde{y}_{\mathrm{c}}(0) = 1\ and by previous inequalities, it follows that:\[\tilde{y}_{\mathrm{c}}(2 k) \geq 2 \left[\tilde{y}_{\mathrm{c}}(k)\right]^2 - 1.\]

Key concepts

Inner Product Space

An inner product space is a vector space equipped with an inner product. The inner product is a generalization of the dot product, which measures angles and lengths in the space.
Using inner product spaces in physics, you can explore many advanced mathematical techniques. It allows us to understand relationships between functions, vectors, and other quantities.
In the given exercise, we use the inner product to work with functions f and g. We define the inner product as \(\textlangle f | g \textrangle\) which provides a way to multiply two functions producing a scalar. For instance, we express the inner product of \(\textlangle h | h \textrangle\) in terms of \(\textlangle f | f \textrangle\), \(\textlangle g | g \textrangle\), and \(\textlangle f | g \textrangle\). This sophistication allows us to derive inequalities and relationships crucial in physics.

Fourier Transform

Fourier transforms are crucial in both mathematics and physics. They convert a time domain function into its frequency domain counterpart. This transformation helps us understand how different frequencies contribute to the overall function.
The Fourier cosine transform, in particular, focuses on the cosine components of a function. In the exercise, the Fourier cosine transform of \(y(x)\) is written as \(\tilde{y}_{\textrm{c}}(k) = \textit\int_{-\infty}^{\infty} y(x) \textit \cos(kx) \textrm{d}x\). Given that \(\tilde{y}_{\textrm{c}}(0)=1\), we use these properties to derive a relationship between \(\tilde{y}_{\text {c}}(2k) \) and \(\tilde{y}_{\text {c}}(k)\).

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a fundamental result in linear algebra and analysis. It is written as \(|\textlangle f | g \textrangle|^2 \leq \textlangle f | f \textrangle \textlangle g | g \textrangle\). It shows how the inner product of two vectors (or functions) is limited by the product of their norms.
In our exercise, we use the positivity of \(\textlangle h | h \textrangle\) to discover that the quadratic expression in \(\textlambda\) must be non-negative, ensuring that its discriminant is zero or negative. This is a key step to connecting it back to the Cauchy-Schwarz inequality, helping us to derive various essential relationships between the inner products and leading to the desired inequality.

Quadratic Expressions

Quadratic expressions are often encountered in physics and mathematics. A typical quadratic expression has the general form \(ax^2 + bx + c\). Analyzing whether such expressions are always non-negative can lead to vital results in applied problems.
In our exercise, we express \(\textlangle h | h \textrangle\) in terms of \(\textlangle f \textrangle\), \(\textlangle g \textrangle\), and \(\lambda\). This derived quadratic expression must always be non-negative for all values of \(\textlambda\). Therefore, we calculate its discriminant and set constraints to ensure it's non-negative. This step allows us to connect several properties, ultimately proving the desired inequality.
The importance of quadratic expressions in this context cannot be overstated, as they form the bridge between abstract inner product properties and practical inequalities in functions.